Page
of 5
ZOOM
Dr. Valerie R. Bencivenga
Economics
329
July 12
, 2018
MIDTERM EXAM #2
Instructions:
Answer the questions below in a blue exam book. There
are
9
questions worth
3
1
0
points. This
is a closed book exam. You may use a calculator (no device with wireless). The formula sheet
and probability
table are
at the end.
Show your work to receive credit.
The exam will last two hours.
Good luck!
(2
5
points)
1.
A small transportation company in the informal sector of Caracas moves construction materials. Below
is the probability dis
tribution of the number of jobs the company gets per day
(X)
:
X
1
2
3
4
P
X
(x)
0
.2
0
.3
0
.4
0
.1
a.
Give the expected number of jobs per day.
b.
Give the variance of the number of jobs per day.
c.
The company charges
$25
per job
.
It pays the driver
$20
per day
(regardless of the number of
jobs).
A
ssume for simplicity that the driver is the company’s only cost.
Write net revenue per day
(R)
as a linear transformation of jobs per day (X). (Net revenue is revenue minus costs.)
d.
What is expected
net revenue per day
?
e
.
What is the variance of net revenue per day?
(
15
points)
2
.
A bicycle shop has a sales business and a repair business. Let
X
represent net revenue
per month
from
the sales business. Expected net revenue
per month
from sales is
8
(
thousand
dollars)
and the variance
of net revenue
per month
from sales is
1.2
(thousand dollars
, squared)
. Let
Y
represent
net
revenue
per
month
from the repair business. Expected
net
revenue
per month
from repairs is
5
(
thousa
n
d dollars)
and the variance of
net
revenue
per month
from repairs is
0.64
(thousand dollars, squared).
The
covariance between
X
and
Y
is
–
0.2.
The shop has fixed costs every month equal to
9
(
thousand
dollars).
(Fixed costs are
rent, utilities,
insurance, wages
, etc., i.e. costs incurred regardless of how much net
revenue there is from the sales business and repair business.)
a.
Let
W
be profits
per month, in thousands of dollars
. Write
W
as a linear combination of
X
and
Y
.
b.
What are expecte
d monthly profits?
c.
What is the variance of monthly profits?
NAME: _____________________________________ EID: ___________________
You
must
return the exam questions along with your blue exam book, or you will
get a zero on the exam. Only your blue exam book will be graded.
(
40
points)
3
.
Here is the bivariate probability distribution of the prices of two stocks
.
X = price of stock #1
10
15
20
100
0
.1
0
.1
0
.1
Y = price of stock #2
200
0
.1
0
.2
0
.1
300
0
.1
0
.1
0
.1
a.
Calculate the expected price (mean) of
X,
and the expected price of
Y. Show your calculations.
(4 points)
b
.
Calculate
the covariance between
X
and
Y.
Show your calculation.
(10 points)
c
.
Are the prices of these two stocks independent?
Justify your answer
.
(6 points)
d
.
Give the conditional probability distribution of
Y
given that
the price of stock #1
(X)
is
20.
(4 points)
e
.
Compute
the expected price of stock #2
(mean of
Y
)
given
that
the price of stock #1
(X)
is
20
.
Are
the unconditional mean of
Y
from part a and the conditional mean of
Y
given
X = 20
the same?
(6 points)
f
.
Compute
the variance of
the
price of stock #2 (variance of
Y
).
(4 points)
g
.
Compute
the variance of the price of stock #2 (the variance of
Y
)
given that
the price of stock #1
(X)
is
20.
Are the unconditional variance of
Y
from part f and the conditional mean of
Y
given
X = 20
the same?
(6 points)
(
40
points)
4
.
A company submits
10
proposals. The probability any one proposal will lead to a contract
is
0
.
3.
The
outcomes of p
roposals are independen
t (“outcome
of a proposal
” means whether or not
the
proposal
leads to a contract)
.
a.
Let
X
represent the number of contracts the company obtains.
What is the expected number of
contracts the company will obtain from its
10
proposals?
What is t
he vari
ance of the number of
contracts?
(6 points)
b.
What is the probability that the company will
obtain at least
2
contracts?
(6 points)
c
.
The company has fixed costs of
400,000.
(Fixed costs do not depend on the number of
contracts or the number of proposals.)
E
ach proposal costs
100,000
to prepare.
Each
contract
generates
500,000
in net reven
ue.
Let
n
represent the number of proposals the company submits. Let
Y
represent the company’s
profits.
Write
p
rofit
(Y)
as a function of
the number of proposals
(n)
and
the number of
contracts
(X).
(10 points)
d.
If the company submits
10
proposals, calculate
expected profit and the variance of profit
.
(8 points)
e
.
What is the
smallest
number of proposals the company can submit yet still have positive
expected
profit?
(10 points)
(30 points)
5.
A company produces
sheets of Kevlar
fabric, for making bullet

proof vests. The company produces
10,000 square meters of fabric per day. The fabric is inspected for defects. Defects are randomly
located on the sheets of fabric. On average, there is
on
e
defect
per 100,000 square meters of f
abric.
a
.
What is the
expected number of
defects
in the
fabric produced
on
one day
?
b
.
What is the probability of
there being
at least one defect
in the
fabric produced
on
one day
?
c
.
What is the
expected number of
defects
in the
fabric produced
over a
three

day
period of time
?
d
.
What is the probability of finding 3
defects
in the
fabric produced
over a
three

day
period of
time
?
e
.
What is the probability of finding one
defect
in the
fabric produced
each day
, for
three days in a
row
?
f
.
Why are t
he
answers to parts d and e different from each other?
(
2
0 points)
6
.
Do
not
calculate your
answers.
A loan officer of a bank in a
village
in India
r
eceive
s
f
ifteen
loan
applications from farmers
,
and five
loan applications
from small businesses
.
The loan officer
has
funds to make nine loans.
T
he loan
off
i
cer chooses nine applic
ations randomly to receive the loans.
a.
W
hat is the probability that a majority of the loans are to small businesses?
b.
What is the probability that
a majority of the loans are to farmers?
c.
W
hat is the probability that
all of the loans are to
farmers?
d.
W
hat is the probability that
all of the loans are to small businesses?
(
5
0
points)
7
.
There are two phases of building a hotel. In phase one, the structure is constructed. In phase two, the
interior is finished (wallpaper, flooring, light fixtures, etc.).
Phase two f
ollows phase one.
Let
X
represent the number of months it takes phase one to be completed. Let
Y
represent the
number of months it takes phase two to be completed. Assume all months have the same number of
work days, and that it can take any fraction of one or more months to complete a phase.
X
is uniformly
distributed between
20
and
30,
and
Y
is u
niformly distributed between
2
0
and
30
.
X
and
Y
are
independent.
a.
State and graph the p.d.f. of
X.
State and graph the c.d.f. of
X.
(
16
points)
b.
Give the expected
value of
X.
Give the variance of
X.
(
6
points)
c.
State the joint p.d.f of
X
and
Y.
Briefly explain or show how you obtained your answer.
(
8
points)
d.
What is the probability that both
X
and
Y
will be greater than
2
5
?
(10 points)
e.
What is the probability that
the total number of months it takes to build the hotel will be
greater than
5
0
?
Let
T
represent the total number of months.
(10 points)
(
40
points)
8
.
For PhD students in economics, lengths of time it takes to complete their dissertations once they have
passed their qualifying exam (“time to completion”) are approximately normally distributed. Thus, “time
to completion” for a randomly

chosen economics PhD student is approximately a normal random
variable
(X).
Assume
X
~ N(31,7
2
)
where
X
is measured in months and a dissertation can take any
fraction of a month to complete.
a.
What is the probability that a randomly

chosen student will
take more than 28 months to
complete their dissertation
?
(5 points)
b
.
What number of months to comple
tion cuts off the 5% of students ta
king the longest to complete
their dissertations? F
ind the number of months
(which can be fractional)
such that 5% of
students tak
e longer than this to complete.
(5 points)
c
.
What is the probability that a student’s t
ime to completion will
be within one month of the
mean
?
(5 points)
d.
In a PhD
class, 25 students h
ave just passed their qualifying exam
. What is the probability that
the average
(mean)
time to completion in this
class
will be
more than 28 months? Assu
me that
these students are a random sample and that “time to completion” is independent across
students
.
(5 points)
e.
What is the probability that the average
(mean)
time to completion in this
class
of 25 students
will be within one month of the populat
ion mean?
(5 points)
f.
The probability you calculated in part d is larger than the probability you calculated in part a. The
probability you calculated in part e is larger than the probability you calculated in part c. Briefly
explain
why
.
(5 points)
g
.
How large
would a class
have to be to ensure that the probability is
at least 0
.9 that the average
time to completion is within one month of the population mean?
(10 points)
Keep going! Question 9 is on the next page.
(
50
points)
9
.
You
plan to
open a small business
in a resort town
. You are deciding whether to
produce
fudge
or
taffy
or
some of each
.
Let
X
represent
t
he number of
pounds of fudge
you
could
make and
sell per week
,
if you
put all of your
time into producing
fudge
.
Assume
X ~ N(
40,
16
).
Let
Y
represent t
he number of
pounds of taffy
you
could
make and
sell per week
,
if you
put all of your
time into producing
taff
y
.
Assume
Y ~ N(
6
0, 25).
The covariance between
X
and
Y
is
negative.
XY
10
(
squared pounds
).
You estimate that net revenue (selling price minus cost of production)
would be
12
dollars
per
pound
of
fudge, and
8
dollars
per
pound
of taffy
.
a.
Use
RX
to represent net revenue of your b
usiness
if you
produce
only
fudge
.
Write
RX
as a
function of
X.
What is expected net revenue?
What is the variance of net revenue?
(
6
points)
b.
What is the probability that
RX
would be
less than
450?
Greater than
510?
(
10
points)
c
.
Use
R
Y
to represent net revenue of your business
if you
produce
only
taff
y
.
Write
RY
as a
function of
Y.
What is expected net revenue? What is the variance of net revenue?
(
6
points)
d.
What is the probability that
RY
would be less than
450?
Greater than
510?
(
10
points)
You could divide your time between producing fudge
and producing taffy. If you spend half of your time
producing fudge, you can make and sell half as much (compared to producing only fudge), and similarly
for taffy.
e
.
Use
RD
to represent net
revenue of your business if you put
half
of your time into
pro
ducing
fudge
, and
half
of your time into
producing
taff
y
.
Write
RD
as a function of
X
and
Y.
What is
expected net revenue? What is the variance of net revenue?
(
6
points)
f
.
What is
the probability that
RD
would
be less than
450?
Greater than
510?
(
10
points)
g.
Which business plan (“only fudge”, “only taffy”, or “half of your time in each”) should you adopt if
your objective is to choose the one with the highest probability of net revenue greater than
510?
No additional calculations are necessary.
(1 point)
Which business play should you adopt if your objective is to choose the one with the smallest
probability of net revenue less than
450? No additional calculations are necessary. (1 point)